Triangulated categories of 1-motivic sheaves
Friday, 30.10.15, 10:15-11:15, Raum 404, Eckerstr. 1
Thanks to the work of Voevodsky, Morel, Ayoub, Cisinski and Déglise, we have at our disposal a mature theory of triangulated categories of mixed motivic sheaves with rational coefficients over general base schemes, with a "six operations" formalism and the expected relationship with algebraic cycles and algebraic K-theory. A parallel development has taken place in the study of Voevodsky's category of mixed motives over a perfect field, where the subcategory of 1-motives (i.e., generated by motives of curves) has been completely described by work of Orgogozo, Barbieri-Viale, Kahn and Ayoub. We explain how to combine these two sets of ideas to study the triangulated category of 1-motivic sheaves over a base. Our main results are the definition of the motivic t-structure for 1-motivic sheaves, a precise relation with Deligne 1-motives, and the extraction of the "1-motivic part" of a general motivic sheaves via a "motivic Picard functor".
Foam categories from categorified quantum groups
Friday, 6.11.15, 10:15-11:15, Hörsaal II, Albertstr. 23b
About 15 years ago, Khovanov introduced an homological invariant of knots categoryfying the Jones polynomial. Though this polynomial can be viewed both from a representation-theoretic and a diagrammatic point of view, for long only the latter version had been categorified.\nI will explain how, inspired by the concept of skew-Howe duality developed by Cautis, Kamnitzer, Morrison and Licata, one can describe the cobordism categories used in Khovanov homology from categorified quantum groups. In turn, this method allowed us to precisely redefine the sln generalizations of these categories, yielding a combinatorial and integral description of Khovanov-Rozansky homologies.
Foam categories from categorified quantum groups
Friday, 6.11.15, 10:15-11:15, Hörsaal II, Albertstr. 23b
About 15 years ago, Khovanov introduced an homological invariant of knots categoryfying the Jones polynomial. Though this polynomial can be viewed both from a representation-theoretic and a diagrammatic point of view, for long only the latter version had been categorified.\nI will explain how, inspired by the concept of skew-Howe duality developed by Cautis, Kamnitzer, Morrison and Licata, one can describe the cobordism categories used in Khovanov homology from categorified quantum groups. In turn, this method allowed us to precisely redefine the sln generalizations of these categories, yielding a combinatorial and integral description of Khovanov-Rozansky homologies.
New counterexamples to Quillen's conjecture
Friday, 13.11.15, 10:15-11:15, Raum 404, Eckerstr. 1
In the talk I will explain the computation of cohomology of \(GL_3\) over function rings of affine elliptic curves. The computation is based on the study of the action of the group on its associated Bruhat-Tits building. It turns out that the equivariant cell structure can be described in terms of a graph of moduli spaces of low-rank vector bundles on the corresponding complete curve. The resulting spectral sequence computation of group cohomology provides very explicit counterexamples to Quillen's conjecture. I will also discuss a possible reformulation of the conjecture using a suitable rank filtration.
Six operations on dg enhancements of derived categories of sheaves and applications
Friday, 20.11.15, 10:15-11:15, Raum 404, Eckerstr. 1
We lift Grothendieck’s six functor formalism for derived categories of sheaves on ringed spaces over a field to differential graded enhancements. If time permits we give applications concerning homological smoothness of derived categories of schemes.
The arc space of Grassmannians
Friday, 27.11.15, 10:00-11:00, Raum 404, Eckerstr. 1
Arc spaces can be used as an effective tool to compute invariants of\nsingularities of algebraic varieties. In this talk, I will explain how this can\nbe achieved for a classical example: the singularities of Schubert varieties\ninside the Grassmannian. This involves a delicate study of the combinatorics\ninside of the arc space of the Grassmannian. The main tool I will discuss is a\nstratification of the arc space which plays the role of a Schubert cell\ndecomposition for lattices. Analyzing the geometric structure of the resulting\nstrata leads to the computation of invariants, mainly the log canonical\nthreshold of pairs invoving Shubert varieties.
Schottky groups acting on homogeneous rational manifolds
Friday, 4.12.15, 10:05-11:05, Raum 404, Eckerstr. 1
In 1877 Schottky constructed free and proper actions of a free group of rank r on a domain in the Riemann sphere having as quotient a compact Riemann surface of genus r. In 1984 Nori extended this construction to any complex-projective space of odd dimension in order to produce compact complex manifolds having free fundamental group. Larusson as well as Seade and Verjovsky studied further properties of these quotient manifolds such as their algebraic and Kodaira dimensions and their deformation theory. In my talk I will explain a joint work with Karl Oeljeklaus where we have studied the question to which homogeneous rational manifolds Nori's construction can be generalized, and the new examples we have found. If time permits, I will also indicate what we can say about geometric and analytic properties of the quotient manifolds associated with these new examples.
Curves of Genus 2 with Bad Reduction and Complex Multiplication
Friday, 11.12.15, 10:00-11:00, Raum 404, Eckerstr. 1
If a smooth projective curve of positive genus which is defined over a number field has good reduction at some finite place, than so does its jacobian. But the converse already fails in genus 2. To study the extent of this failure we investigate jacobians that have complex multiplication. This forces the jacobians to have potentially good reduction at all finite places by a theorem of Serre and Tate. I will present a result which roughly speaking states that a genus 2 curve whose jacobian has complex multiplication usually has bad stable reduction at at least one finite place. This is joint work with Fabien Pazuki.
On Beauville's conjectural weak splitting property
Friday, 18.12.15, 10:15-11:15, Raum 404, Eckerstr. 1
We present a result on the Chow ring of irreducible symplectic varieties. The main object of interest is Beauville's conjectural weak splitting property, which predicts the injectivity of the cycle class map restricted to a certain subalgebra of the rational Chow ring (the subalgebra generated by divisor classes). For special irreducible symplectic varieties we relate it to a conjecture on the existence of rational Lagrangian fibrations. After deducing that this implies the weak splitting property in many new cases, we present parts of the proof.
Non-archimedean links of singularities
Friday, 8.1.16, 10:15-11:15, Raum 404, Eckerstr. 1
I will introduce a non-archimedean version of the link of a singularity. This object will be a space of valuations, a close relative of non-archimedean analytic spaces (in the sense of Berkovich) over trivially valued fields. \nAfter describing the structure of these links, I will deduce information about the resolutions of surface singularities. \nIf times allows, I will then characterize those normal surface singularities whose link satisfies a self-similarity property. The last part is a current work in progress with Charles Favre and Matteo Ruggiero.
Classifying line bundles over rigid analytic varieties
Friday, 15.1.16, 10:15-11:15, Raum 404, Eckerstr. 1
Degenerations of polarized Calabi-Yau manifolds
Friday, 22.1.16, 10:00-11:00, Raum 404, Eckerstr. 1
I will present a joint work with Mattias Jonsson in which we rely on non-Archimedean geometry to study the limit behavior of the volume forms of Ricci-flat Kähler metrics in a degenerating family.
Frobenius splittings in birational geometry
Friday, 29.1.16, 10:15-11:15, Raum 404, Eckerstr. 1
Frobenius splittings in birational geometry
Friday, 29.1.16, 10:15-11:15, Raum 404, Eckerstr. 1
Due to the absence of the Kawamata-Viehweg vanishing theorem, the classification of algebraic varieties in positive characteristic, as of very recently, has been seen as an insurmountable task. Recent progress in the field has been inspired by the discovery of Frobenius-split varieties. In my talk, I will discuss connections between the geometry of projective varieties and properties of the Frobenius action, focusing particularly on surfaces.
The cotangential and the derived de Rham complex in the h-topology
Tuesday, 2.2.16, 10:15-11:15, Raum 404, Eckerstr. 1
Friday, 5.2.16, 10:00-11:00, Raum 404, Eckerstr. 1
Abundance conjecture for varieties with many differential forms
Friday, 5.2.16, 10:00-11:00, Raum 404, Eckerstr. 1
The abundance conjecture and the existence of good models are the main open problems in the Minimal Model Program in complex algebraic geometry. Even though it is completely proved in dimension 3, almost nothing has been known in higher dimensions. In this talk, I will discuss my recent joint work with Thomas Peternell, where we prove that the abundance conjecture holds on a variety with mild singularities if it has many reflexive differential forms with coeffi cients in pluricanonical bundles, assuming the Minimal Model Program in lower dimensions. Under this assumption, the result has several consequences: for instance, that hermitian semipositive canonical divisors are almost always semiample. When the numerical dimension of the canonical sheaf is 1, our results hold unconditionally in every dimension.
Effective Matsusaka for surfaces in positive characteristic
Friday, 12.2.16, 10:00-11:00, Raum 404, Eckerstr. 1
The problem of determining an effective bound on the multiple which makes an ample divisor D on a smooth variety X very ample is natural and many results are known in characteristic zero. In this talk, based on a joint paper with Gabriele Di Cerbo, I will discuss this problem on surfaces in positive characteristic, giving a complete solution in this setting. \nOur strategy requires an ad hoc study of pathological surfaces, on which Kodaira-type theorems can fail. A Fujita-type theorem and a vanishing result for big and nef divisors on pathological surfaces will also be discussed.
Etale motivic cohomology
Friday, 11.3.16, 10:15-11:15, Raum 404, Eckerstr. 1
We discuss structure theorems and duality\nresults for etale integral motivic cohomology\nof smooth projective varieties over algebraically closed,\nfinite, and local fields.\n\n